Mutations in Fallout 76 are a type of condition that typically gives you one buff and one nerf at the same time, making your character better at certain things, and worse at other things. They are non-permanent options, so you can always remove them by using a RadAway if you don't want the mutation anymore, so it's not really a problem if you end up with a "bad" mutation. The mutations persists through death, but will be removed by sources that also heals radiation damage.
Anyway, you get mutations when you are exposed to radiation. You have a 5% chance to get a mutation per 5HP taken from radioactive sources. This include sources like drinking unfiltered water, being close to nuclear waste etc.
This means that you will typically need to be exposed to over 100 HP worth of radiation damage before you get a mutation, even tough you can get lucky and get one earlier.
You always get a random mutation when you get mutations this way, and there are currently 19 different mutations that can affect your character. You can have multiple mutations affecting your character at the same time, but some of them are mutually exclusive, so you will not be able to get all 19 mutations at the same time.